As adults, we have to work with money every day. While balancing your checkbook or calculating your monthly expenditures only requires simple arithmetic, when we start saving, planning for retirement, or need a loan, we need to incorporate some higher mathematical skills.
Discussing interest starts with the principal, the amount your account starts with, or even the initial amount of money borrowed. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed \$100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of \$100 which is \$100 × 0.05 = \$5. The total amount you would pay your friend back would be \$105, the original principal plus the interest that accrued.
$P_0$: is the initial amount (principal)
$A$: is the future value
$t$: is the length of the loan in years
$r$: interest rate as a decimal
$I$: is the amount of interest accrued over $t$ years
What makes Simple Interest different from its counterparts is that it is calculated as strictly a percentage of the original amount. Meaning that regardless of the interest that is accrued over the years, the interest that is gained will only be based on that of the initial investment.
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
$P_0 = \$300$ (Principal)
$r = 0.03$ (3% Interest Rate)
Using $I = P_0 r t$ we are assuming $t = 1$ in this case as this is a one-time interest application.
$$I = \$300 \times 0.03 = \$9.00$$This means when your friend goes to pay you back, they owe you \$9.00 extra dollars in interest. This means they will pay you $\$300 + \$9.00 = \$309$
One-time simple interest is only common for extremely short-term loans (typically less than a year). For longer term loans, it is common for interest to be paid on more of an annual basis. For example, bonds are essentially a loan made to the bond issuer (a company or the government) by you, the bond holder. In return for the loan, the issuer agreed to pay interest. Bonds have a maturity date, at which time the issuer pays back the original bond value, plus the interest that accrued over the life of the bond.
Suppose your city is building a new park and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that will mature in 5 years. How much interest will you earn?
$P_0 = \$1,000$ (Principal)
$r = 0.05$ (5% Interest Rate)
$t = 5$ (5 years)
Using $I = P_0 r t$:
$$I = \$1,000 \times 0.05 \times 5 = \$250$$Thus when the bond matures, the city will pay you back your original \$1,000 plus \$250 that you gained in interest for a future value of $\$1,000 + \$250 = \$1,250$.
Interest rates are usually given as an annual percentage rate (APR) - the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided into the number of increments.
Treasury Notes (T-Notes) are bonds issued by the federal government to cover their expenses. Suppose you obtain a \$1,000 T-note with a 4% APR, paid semi-annually, with a maturity in 4 years. How much interest will you have earned?
Since interest is being paid semi-annually (twice a year), the APR of 4% would have to be divided by 2 to get 2%.
$P_0 = \$1,000$ (Principal)
$r = 0.02$ (2% rate per half year)
$t = 8$ (4 years = 8 half-years)
Over the 4 year period, you will have earned $160.
A loan company charges $30 interest for a one month loan of \$500. Find the annual interest rate they are charging.